Extensions of orthosymmetric lattice bimorphisms

Abstract

Abstract. Let E be an Archimedean vector lattice, let E d be its Dedekind completion and let B be a Dedekind complete vector lattice. If Ψ 0 : E × E → B is an orthosymmetric lattice bimorphism, then there exists a lattice bimorphism Ψ :that not just extends Ψ 0 but also has to be orthosymmetric. As an application, we prove the following: Let A be an Archimedean d-algebra. Then the multiplication in A can be extended to a multiplication in A d , the Dedekind completion of A, in such a fashion that A d is again a d-algebra with respect to this extended multiplication. This gives a positive answer to the problem posed by C. B. Huijsmans in 1990.

On extensions of orthosymmetric lattice bimorphisms

“…The aim of the present paper is to indicate the error in the proof of the above mentioned result and to give a correct formulation concerning extension of orthosymmetric lattice bilinear maps. With help of a suitable example we show that ( [13], Theorem 1) and its proof cannot be improved. Apart from that, we are going to investigate the problem of extension of multiplications for d-algebras, which is a question posed by Huijsmans in [8] (last paragraph of section 7): can the multiplication of d-algebra be extended to its Dedekind completion?…”

“…If Ψ 0 : A × A → B is an orthosymmetric lattice bilinear map, then every lattice extension Ψ : A d × A d → B of Ψ 0 is again orthosymmetric. Unfortunately, the result and its proof given in [13] are not correct. The aim of the present paper is to indicate the error in the proof of the above mentioned result and to give a correct formulation concerning extension of orthosymmetric lattice bilinear maps.…”

“…In [13] the author proved the following result: let A be an Archimedean vector lattice, let A d be its Dedekind completion and let B be a Dedekind complete vector lattice. If Ψ 0 : A × A → B is an orthosymmetric lattice bilinear map, then every lattice extension Ψ : A d × A d → B of Ψ 0 is again orthosymmetric.…”

“…Apart from that, we are going to investigate the problem of extension of multiplications for d-algebras, which is a question posed by Huijsmans in [8] (last paragraph of section 7): can the multiplication of d-algebra be extended to its Dedekind completion? Under some assumptions (being a slight modification of the hypotheses imposed in [13]) we prove a result on the extension of orthosymmetric lattice bilinear maps. The main tool used in our proof is the technique associated with extension to a vector subspace generated by adjoining one element.…”